Congruent triangle tests
Congruent triangles have the same size and shape. Match corresponding vertices first, then check whether the information satisfies SSS, SAS, AAS or RHS
- Same size and shape Corresponding sides and angles are equal. Turning or reflecting a triangle does not change its size or shape
- Corresponding order In ABC congruent to DEF, A matches D, B matches E and C matches F. Follow the order, not the position on the page
- SSS Three pairs of corresponding sides are equal
- SAS Two corresponding side pairs and the angle between those sides are equal
- AAS or ASA Two corresponding angle pairs and any corresponding side are equal. The third angle is fixed by the 180° total, so ASA is equivalent
- RHS Both triangles are right-angled, with equal hypotenuses and one other corresponding side equal
- Insufficient data AAA fixes shape but not size. SSA does not generally fix one triangle. Equal area alone is also insufficient
- Explain why a test works SSS fixes the third vertex up to reflection. AAS/ASA fixes the third angle. For RHS, Pythagoras fixes the remaining leg, reducing the reasoning to SSS
- Build and use a proof List the matched facts with reasons, including a common side when needed. Name the test and write vertices in corresponding order. Only then use equal corresponding parts to deduce a further result
- Insufficient does not mean false Measurements that do not establish congruence may still belong to congruent triangles. Say what the given evidence proves; do not infer a mismatch from missing information
Worked example
In triangles ABC and ADC, sides AB and AD are each 7 cm, sides BC and DC are each 9 cm, and AC is common. Prove that the triangles are congruent and hence prove that angle BAC equals angle DAC
- \(\displaystyle AB=AD\)Given equal side lengths
- \(\displaystyle BC=DC\)Given equal side lengths
- \(\displaystyle AC=AC\)Common side
- \(\displaystyle \triangle ABC\cong\triangle ADC\)Three corresponding side pairs are equal, so SSS applies
- \(\displaystyle \angle BAC=\angle DAC\)Corresponding angles of the proved congruent triangles are equal
Common mistakes
Difficulty levels
Questions about this topic
What does SSS mean?+
SSS means side-side-side. Three pairs of equal corresponding side lengths are enough to prove triangle congruence
Why is AAA not a congruence test?+
AAA determines the angles and therefore the shape, but a triangle with the same angles can have a different size
What does the order in a congruence statement show?+
It shows the correspondence between vertices. The positions in the two triangle names must match
Is ASA also a valid test?+
Yes. Two angles determine the third, so an equal corresponding side between those angles works just as it does for AAS
Why does RHS require a hypotenuse?+
The hypotenuse is opposite the right angle. RHS requires that longest side and one other side to match, with both right angles established
Why does RHS work when SSA usually does not?+
For right triangles, the hypotenuse and one leg determine the other leg by Pythagoras. This fixes all three side lengths. Without the right-angle condition, two constructions can fit the same SSA data
What should I write in a complete congruence proof?+
Give three matched facts with valid reasons, name the test, state the triangles in corresponding order, and justify any further result using corresponding parts
Does failing a test prove the triangles are different?+
Not necessarily. Insufficient information leaves the question unsettled. An actual unequal pair of corresponding sides or angles would prove a mismatch
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 7: Geometry and similarity